\(\int (b \sec (e+f x))^{4/3} (d \tan (e+f x))^{3/2} \, dx\) [339]

Optimal result

Integrand size = 25, antiderivative size = 64 \[ \int (b \sec (e+f x))^{4/3} (d \tan (e+f x))^{3/2} \, dx=\frac {2 \cos ^2(e+f x)^{23/12} \operatorname {Hypergeometric2F1}\left (\frac {5}{4},\frac {23}{12},\frac {9}{4},\sin ^2(e+f x)\right ) (b \sec (e+f x))^{4/3} (d \tan (e+f x))^{5/2}}{5 d f} \]

[Out]

2/5*(cos(f*x+e)^2)^(23/12)*hypergeom([5/4, 23/12],[9/4],sin(f*x+e)^2)*(b*sec(f*x+e))^(4/3)*(d*tan(f*x+e))^(5/2
)/d/f

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.040, Rules used = {2697} \[ \int (b \sec (e+f x))^{4/3} (d \tan (e+f x))^{3/2} \, dx=\frac {2 \cos ^2(e+f x)^{23/12} (b \sec (e+f x))^{4/3} (d \tan (e+f x))^{5/2} \operatorname {Hypergeometric2F1}\left (\frac {5}{4},\frac {23}{12},\frac {9}{4},\sin ^2(e+f x)\right )}{5 d f} \]

[In]

Int[(b*Sec[e + f*x])^(4/3)*(d*Tan[e + f*x])^(3/2),x]

[Out]

(2*(Cos[e + f*x]^2)^(23/12)*Hypergeometric2F1[5/4, 23/12, 9/4, Sin[e + f*x]^2]*(b*Sec[e + f*x])^(4/3)*(d*Tan[e
+ f*x])^(5/2))/(5*d*f)

Rule 2697

Int[((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(a*Sec[e + f
*x])^m*(b*Tan[e + f*x])^(n + 1)*((Cos[e + f*x]^2)^((m + n + 1)/2)/(b*f*(n + 1)))*Hypergeometric2F1[(n + 1)/2,
(m + n + 1)/2, (n + 3)/2, Sin[e + f*x]^2], x] /; FreeQ[{a, b, e, f, m, n}, x] &&  !IntegerQ[(n - 1)/2] &&  !In
tegerQ[m/2]

Rubi steps \begin{align*} \text {integral}& = \frac {2 \cos ^2(e+f x)^{23/12} \operatorname {Hypergeometric2F1}\left (\frac {5}{4},\frac {23}{12},\frac {9}{4},\sin ^2(e+f x)\right ) (b \sec (e+f x))^{4/3} (d \tan (e+f x))^{5/2}}{5 d f} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.11 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.00 \[ \int (b \sec (e+f x))^{4/3} (d \tan (e+f x))^{3/2} \, dx=\frac {3 d \operatorname {Hypergeometric2F1}\left (-\frac {1}{4},\frac {2}{3},\frac {5}{3},\sec ^2(e+f x)\right ) (b \sec (e+f x))^{4/3} \sqrt {d \tan (e+f x)}}{4 f \sqrt [4]{-\tan ^2(e+f x)}} \]

[In]

Integrate[(b*Sec[e + f*x])^(4/3)*(d*Tan[e + f*x])^(3/2),x]

[Out]

(3*d*Hypergeometric2F1[-1/4, 2/3, 5/3, Sec[e + f*x]^2]*(b*Sec[e + f*x])^(4/3)*Sqrt[d*Tan[e + f*x]])/(4*f*(-Tan
[e + f*x]^2)^(1/4))

Maple [F]

\[\int \left (b \sec \left (f x +e \right )\right )^{\frac {4}{3}} \left (d \tan \left (f x +e \right )\right )^{\frac {3}{2}}d x\]

[In]

int((b*sec(f*x+e))^(4/3)*(d*tan(f*x+e))^(3/2),x)

[Out]

int((b*sec(f*x+e))^(4/3)*(d*tan(f*x+e))^(3/2),x)

Fricas [F]

\[ \int (b \sec (e+f x))^{4/3} (d \tan (e+f x))^{3/2} \, dx=\int { \left (b \sec \left (f x + e\right )\right )^{\frac {4}{3}} \left (d \tan \left (f x + e\right )\right )^{\frac {3}{2}} \,d x } \]

[In]

integrate((b*sec(f*x+e))^(4/3)*(d*tan(f*x+e))^(3/2),x, algorithm="fricas")

[Out]

integral((b*sec(f*x + e))^(1/3)*sqrt(d*tan(f*x + e))*b*d*sec(f*x + e)*tan(f*x + e), x)

Sympy [F(-1)]

Timed out. \[ \int (b \sec (e+f x))^{4/3} (d \tan (e+f x))^{3/2} \, dx=\text {Timed out} \]

[In]

integrate((b*sec(f*x+e))**(4/3)*(d*tan(f*x+e))**(3/2),x)

[Out]

Timed out

Maxima [F]

\[ \int (b \sec (e+f x))^{4/3} (d \tan (e+f x))^{3/2} \, dx=\int { \left (b \sec \left (f x + e\right )\right )^{\frac {4}{3}} \left (d \tan \left (f x + e\right )\right )^{\frac {3}{2}} \,d x } \]

[In]

integrate((b*sec(f*x+e))^(4/3)*(d*tan(f*x+e))^(3/2),x, algorithm="maxima")

[Out]

integrate((b*sec(f*x + e))^(4/3)*(d*tan(f*x + e))^(3/2), x)

Giac [F]

\[ \int (b \sec (e+f x))^{4/3} (d \tan (e+f x))^{3/2} \, dx=\int { \left (b \sec \left (f x + e\right )\right )^{\frac {4}{3}} \left (d \tan \left (f x + e\right )\right )^{\frac {3}{2}} \,d x } \]

[In]

integrate((b*sec(f*x+e))^(4/3)*(d*tan(f*x+e))^(3/2),x, algorithm="giac")

[Out]

integrate((b*sec(f*x + e))^(4/3)*(d*tan(f*x + e))^(3/2), x)

Mupad [F(-1)]

Timed out. \[ \int (b \sec (e+f x))^{4/3} (d \tan (e+f x))^{3/2} \, dx=\int {\left (d\,\mathrm {tan}\left (e+f\,x\right )\right )}^{3/2}\,{\left (\frac {b}{\cos \left (e+f\,x\right )}\right )}^{4/3} \,d x \]

[In]

int((d*tan(e + f*x))^(3/2)*(b/cos(e + f*x))^(4/3),x)

[Out]

int((d*tan(e + f*x))^(3/2)*(b/cos(e + f*x))^(4/3), x)